# Structural Depth Wave4 Roots: Rings, Metric Completeness, Linear Structures, Independence and Graph Paths Complete seven traceable knowledge objects in dependency order on the existing number-system and function foundations. This is a suggested learning sequence, not a universal logical dependency or a formal proof. - Stage 1 · required ## [Finite abelian group](/en/mko/mko-finite-abelian-group/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261027%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-finite-abelian-group) [Concept guides](/en/derivations/?q=mko-finite-abelian-group) - Stage 1 · required ## [Arithmetic operations](/en/mko/mko-arithmetic-operations/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261027%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-arithmetic-operations) [Concept guides](/en/derivations/?q=mko-arithmetic-operations) - Stage 1 · required ## [Function mapping](/en/mko/mko-function-mapping/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261027%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-function-mapping) [Concept guides](/en/derivations/?q=mko-function-mapping) - Stage 1 · required ## [Real normed vector space](/en/mko/mko-real-normed-vector-space/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261027%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-real-normed-vector-space) [Concept guides](/en/derivations/?q=mko-real-normed-vector-space) - Stage 1 · required ## [Metric space](/en/mko/mko-metric-space/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261027%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-metric-space) [Concept guides](/en/derivations/?q=mko-metric-space) - Stage 1 · required ## [Real vector space](/en/mko/mko-real-vector-space/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261027%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-real-vector-space) [Concept guides](/en/derivations/?q=mko-real-vector-space) - Stage 1 · required ## [Cauchy sequence](/en/mko/mko-cauchy-sequence/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261027%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-cauchy-sequence) [Concept guides](/en/derivations/?q=mko-cauchy-sequence) - Stage 1 · required ## [Convergence in a metric space](/en/mko/mko-metric-convergence/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261027%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-metric-convergence) [Concept guides](/en/derivations/?q=mko-metric-convergence) - Stage 1 · required ## [Linear combinations and span](/en/mko/mko-linear-combination-span/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261027%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-linear-combination-span) [Concept guides](/en/derivations/?q=mko-linear-combination-span) - Stage 1 · required ## [Event](/en/mko/mko-event/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261027%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-event) [Concept guides](/en/derivations/?q=mko-event) - Stage 1 · required ## [Probability on finite sample spaces](/en/mko/mko-probability/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261027%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-probability) [Concept guides](/en/derivations/?q=mko-probability) - Stage 1 · required ## [Conditional probability in finite models](/en/mko/mko-conditional-probability/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261027%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-conditional-probability) [Concept guides](/en/derivations/?q=mko-conditional-probability) - Stage 1 · required ## [Finite simple undirected graphs](/en/mko/mko-graph/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261027%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-graph) [Concept guides](/en/derivations/?q=mko-graph) - Stage 1 · required ## [Sequence](/en/mko/mko-sequence/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261027%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-sequence) [Concept guides](/en/derivations/?q=mko-sequence) - Stage 1 · required ## [Binary relations and set projections](/en/mko/mko-relation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261027%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-relation) [Concept guides](/en/derivations/?q=mko-relation) - Stage 2 · optional ## [Finite unital ring](/en/mko/mko-finite-unital-ring/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261027%2F) On one complete finite carrier, this unital-ring protocol requires an additive abelian group, associative multiplication, both distributive laws and a multiplicative identity different from zero. [Related questions](/en/explore/?mko=mko-finite-unital-ring) [Concept guides](/en/derivations/?q=mko-finite-unital-ring) - Stage 3 · optional ## [Norm-induced metric](/en/mko/mko-norm-induced-metric/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261027%2F) On the same real normed vector space, taking the norm of a vector difference defines a metric whose axioms follow universally from the norm axioms. [Related questions](/en/explore/?mko=mko-norm-induced-metric) [Concept guides](/en/derivations/?q=mko-norm-induced-metric) - Stage 4 · optional ## [Complete metric space](/en/mko/mko-complete-metric-space/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261027%2F) A metric space is complete when every Cauchy sequence converges to a point still belonging to that space. [Related questions](/en/explore/?mko=mko-complete-metric-space) [Concept guides](/en/derivations/?q=mko-complete-metric-space) - Stage 5 · optional ## [Real bilinear form](/en/mko/mko-real-bilinear-form/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261027%2F) A bilinear form on a real vector space preserves addition and real scalar multiplication separately in each input. [Related questions](/en/explore/?mko=mko-real-bilinear-form) [Concept guides](/en/derivations/?q=mko-real-bilinear-form) - Stage 6 · optional ## [Linear map between real vector spaces](/en/mko/mko-general-linear-map/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261027%2F) A map between real vector spaces is linear when it preserves linear combinations for all vectors and real scalars. [Related questions](/en/explore/?mko=mko-general-linear-map) [Concept guides](/en/derivations/?q=mko-general-linear-map) - Stage 7 · optional ## [Event independence](/en/mko/mko-event-independence/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261027%2F) Two events are independent when their intersection probability equals the product of their marginal probabilities; this definition does not rely on conditioning on a zero-probability event. [Related questions](/en/explore/?mko=mko-event-independence) [Concept guides](/en/derivations/?q=mko-event-independence) - Stage 8 · optional ## [Graph walks and simple paths](/en/mko/mko-graph-walk-simple-path/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261027%2F) In a finite undirected simple graph, a walk requires each consecutive pair of vertices to be adjacent; a simple path additionally requires all vertices to be distinct. [Related questions](/en/explore/?mko=mko-graph-walk-simple-path) [Concept guides](/en/derivations/?q=mko-graph-walk-simple-path) Path source & statuspath-pulse-foundations-20261027 · candidate [Original paths JSON](/data/architecture/learning-paths.json)